Implement dual subdivision and weight vectors for tropical variety #38536

verreld7 · 2024-08-20T13:56:28Z

This PR introduces some new methods in TropicalVariety related to dual subdivision and weight vectors.

Summary of changes:

Add dual_subdivision in TropicalVariety: return the dual subdivision for any tropical variety as a Graph object
Add _components_of_vertices in TropicalCurve: intermediate method the help calculate weight vectors
Add weight_vectors in TropicalCurve: return the weight vectors of each vertex
Add weight_vectors in TropicalSurface: return the weight vectors of each unique line of intersection
Add a few methods related to dual subdivision of tropical curve such as:
- is_smooth
- is_simple
- genus
- contribution

CC: @tscrim

📝 Checklist

The title is concise and informative.
The description explains in detail what this PR is about.
I have linked a relevant issue or discussion.
I have created tests covering the changes.
I have updated the documentation and checked the documentation preview.

⌛ Dependencies

#37962: issues related to this PR
#38291: continuation of this PR

github-actions · 2024-08-22T06:42:35Z

Documentation preview for this PR (built with commit 204a525; changes) is ready! 🎉
This preview will update shortly after each push to this PR.

tscrim

Sorry for taking so long to get to this. Here are my comments.

tscrim · 2024-10-10T00:34:44Z

src/sage/rings/semirings/tropical_mpolynomial.py

@@ -390,6 +405,180 @@ def tropical_variety(self):
            return TropicalSurface(self)
        return TropicalVariety(self)

+    def Newton_polytope(self):


Suggested change

def Newton_polytope(self):

def newton_polytope(self):

We follow Python's naming conventions, even when the words are proper nouns.

tscrim · 2024-10-10T00:36:01Z

src/sage/rings/semirings/tropical_mpolynomial.py

+            sage: p3 = R(8) + R(4)*x + R(2)*y + R(1)*x^2 + x*y + R(1)*y^2 \
+            ....:     + R(2)*x^3 + x^2*y + x*y^2 + R(4)*y^3 + R(8)*x^4 \
+            ....:     + R(4)*x^3*y + x^2*y^2 + R(2)*x*y^3 + y^4


Suggested change

sage: p3 = R(8) + R(4)*x + R(2)*y + R(1)*x^2 + x*y + R(1)*y^2 \

....: + R(2)*x^3 + x^2*y + x*y^2 + R(4)*y^3 + R(8)*x^4 \

....: + R(4)*x^3*y + x^2*y^2 + R(2)*x*y^3 + y^4

sage: p3 = (R(8) + R(4)*x + R(2)*y + R(1)*x^2 + x*y + R(1)*y^2

....: + R(2)*x^3 + x^2*y + x*y^2 + R(4)*y^3 + R(8)*x^4

....: + R(4)*x^3*y + x^2*y^2 + R(2)*x*y^3 + y^4)

You can see in the built html doc the output is strange. Plus this is considered to be better practice.

tscrim · 2024-10-10T00:36:33Z

src/sage/rings/semirings/tropical_mpolynomial.py

+            p3 = R(8) + R(4)*x + R(2)*y + R(1)*x**2 + x*y + R(1)*y**2 \
+                 + R(2)*x**3 + x**2*y + x*y**2 + R(4)*y**3 + R(8)*x**4 \
+                 + R(4)*x**3*y + x**2*y**2 + R(2)*x*y**3 + y**4
+            sphinx_plot(p3.dual_subdivision().plot())


Suggested change

p3 = R(8) + R(4)*x + R(2)*y + R(1)*x**2 + x*y + R(1)*y**2 \

+ R(2)*x**3 + x**2*y + x*y**2 + R(4)*y**3 + R(8)*x**4 \

+ R(4)*x**3*y + x**2*y**2 + R(2)*x*y**3 + y**4

sphinx_plot(p3.dual_subdivision().plot())

p3 = (R(8) + R(4)*x + R(2)*y + R(1)*x**2 + x*y + R(1)*y**2

+ R(2)*x**3 + x**2*y + x*y**2 + R(4)*y**3 + R(8)*x**4

+ R(4)*x**3*y + x**2*y**2 + R(2)*x*y**3 + y**4)

sphinx_plot(p3.dual_subdivision().plot())

tscrim · 2024-10-10T00:38:02Z

src/sage/rings/semirings/tropical_polynomial.py

-    sage: (x + R(3)*x) * (x^2 + x)
-    3*x^3 + 3*x^2
-    sage: (x^2 + R(1)*x + R(-1))^2
-    0*x^4 + 1*x^3 + 2*x^2 + 0*x + (-2)
+    sage: (x^2 + x + R(0))^2
+    0*x^4 + 0*x^3 + 0*x^2 + 0*x + 0


Why did you change these tests? Why not just add the extra one?

tscrim · 2024-10-10T00:38:31Z

src/sage/rings/semirings/tropical_polynomial.py

-        OUTPUT: a list of tropical numbers
+        OUTPUT: A list of tropical numbers


Please revert this as our style is (usually) starting with a lower case in this instance.

tscrim · 2024-10-10T00:44:32Z

src/sage/rings/semirings/tropical_mpolynomial.py

+        corresponding to the exponents of the monomials of tropical
+        polynomial.
+
+        OUTPUT: :class:`sage.geometry.polyhedron.constructor.Polyhedron`


Suggested change

OUTPUT: :class:`sage.geometry.polyhedron.constructor.Polyhedron`

OUTPUT: :class:`~sage.geometry.polyhedron.constructor.Polyhedron`

tscrim · 2024-10-10T00:44:43Z

src/sage/rings/semirings/tropical_mpolynomial.py

+        the intersection of multiple components. Edges of the dual
+        subdivision correspond to the individual components.
+
+        OUTPUT: :class:`sage.geometry.polyhedral_complex.PolyhedralComplex`


Suggested change

OUTPUT: :class:`sage.geometry.polyhedral_complex.PolyhedralComplex`

OUTPUT: :class:`~sage.geometry.polyhedral_complex.PolyhedralComplex`

tscrim · 2024-10-10T00:44:51Z

src/sage/rings/semirings/tropical_mpolynomial.py

+            p1 = x + y
+            sphinx_plot(p1.Newton_polytope().plot())
+
+        Newton polytope in three dimension::


Suggested change

Newton polytope in three dimension::

A Newton polytope in three dimension::

tscrim · 2024-10-10T00:44:56Z

src/sage/rings/semirings/tropical_mpolynomial.py

+
+        EXAMPLES:
+
+        Newton polytope for a two-variable tropical polynomial::


Suggested change

Newton polytope for a two-variable tropical polynomial::

A Newton polytope for a two-variable tropical polynomial::

tscrim · 2024-10-10T00:48:07Z

src/sage/rings/semirings/tropical_variety.py

+            sage: tv.weight_vectors()
+            ({0: ((u1, u3 - 7/3, u3 - 10/3, u3), {u1 <= u3 - 22/3}),


The Conda test is reporting finding a slightly different solution. So it might be better to test the properties (that the balancing condition is satisfied) rather than the explicit output. (Although it is better to demonstrate the balancing condition does hold anyways.)

verreld7 added 4 commits August 20, 2024 20:42

Add dual_subdivision

06841b5

Add _components_of_vertices and weight_vectors

6734319

Add is_smooth, is_simple, genus, contribution

8b8c5cc

Add weight_vectors in tropical surface

5c9e8d1

verreld7 added 8 commits August 22, 2024 22:43

Add PLOT section in dual subdvision and add one reference

010149c

Add TESTS to genus and contribution

3f1641a

Small fix on style and white spaces

4795639

Refactor vertices in tropical curve

a601cbd

Move weight_vectors from TropicalSurface to TropicalVariety

2e8592c

Small fix on example

96a0f4c

Another fix

ec4307d

Fix docstring in weight_vectors

401ad51

tscrim self-requested a review August 27, 2024 05:41

verreld7 added 10 commits August 30, 2024 17:08

Change output of dual subdivision from Graph to PolyhedralComplex

792b829

Refine docstring

dd2d14d

Small fix on example

f20838a

Refine the docstring more

f425552

Add more examples to TropicalCurve

4306750

Move dual subdivision to TropicalMPolynomial

3d05257

Add polytope to TropicalMPolynomial

d2d211b

Fix PLOT section in dual subdivision

53f495e

Fix dual subdivision

9f66327

Modify dual subdivisio to utilize weight vectors

2fe9f38

tscrim added c: algebraic geometry c: commutative algebra t: enhancement labels Sep 10, 2024

verreld7 added 3 commits September 15, 2024 15:58

Merge branch 'sagemath:develop' into dualsubdivision

b07189d

Add more examples for dual subdivision

87cd7bc

Merge branch 'develop' into dualsubdivision

d4c1251

Fix lint error

204a525

verreld7 marked this pull request as ready for review September 25, 2024 12:34

github-actions bot added the s: needs review label Sep 25, 2024

tscrim requested changes Oct 10, 2024

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Implement dual subdivision and weight vectors for tropical variety #38536

Implement dual subdivision and weight vectors for tropical variety #38536

verreld7 commented Aug 20, 2024 •

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tscrim Oct 10, 2024

tscrim Oct 10, 2024

tscrim Oct 10, 2024

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tscrim Oct 10, 2024

		OUTPUT: a list of tropical numbers
		OUTPUT: A list of tropical numbers

	OUTPUT: :class:`sage.geometry.polyhedron.constructor.Polyhedron`
	OUTPUT: :class:`~sage.geometry.polyhedron.constructor.Polyhedron`

	OUTPUT: :class:`sage.geometry.polyhedral_complex.PolyhedralComplex`
	OUTPUT: :class:`~sage.geometry.polyhedral_complex.PolyhedralComplex`

	Newton polytope in three dimension::
	A Newton polytope in three dimension::


		EXAMPLES:

		Newton polytope for a two-variable tropical polynomial::

	Newton polytope for a two-variable tropical polynomial::
	A Newton polytope for a two-variable tropical polynomial::

		sage: tv.weight_vectors()
		({0: ((u1, u3 - 7/3, u3 - 10/3, u3), {u1 <= u3 - 22/3}),

Implement dual subdivision and weight vectors for tropical variety #38536

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Implement dual subdivision and weight vectors for tropical variety #38536

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verreld7 commented Aug 20, 2024 • edited Loading

Summary of changes:

📝 Checklist

⌛ Dependencies

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verreld7 commented Aug 20, 2024 •

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github-actions bot commented Aug 22, 2024 •

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